Bombieri-Vinogradov theorem

The Bombieri-Vinogradov theorem, sometimes called Bombieri’s theorem, states that for a positive real number $A$, if $x^{\frac{1}{2}}{\log }^{-A}x\le Q\le x^{\frac{1}{2}}$ then

$$\sum _{q\le Q}\underset{y\le x}{\max }\underset{\genfrac{}{}{0pt}{}{1\le a\le q}{(a,q)=1}}{\max }\left|\psi (x;q,a)-\frac{x}{\varphi (q)}\right|=O\left(x^{\frac{1}{2}}Q{(\log x)}^5\right),$$

where $\varphi (q)$ is Euler’s totient function and

$$\psi (x;q,a)=\sum _{\genfrac{}{}{0pt}{}{n\le x}{n\equiv amodq}}\mathrm{\Lambda }(n),$$

where $\mathrm{\Lambda }(n)$ is the Mangoldt function.